Solving the Faddeev-Merkuriev Equations in Total Orbital Momentum Representation via Spline Collocation and Tensor Product Preconditioning

Abstract

The computational approach for solving the Faddeev-Merkuriev equations in total orbital momentum representation is presented. These equations describe a system of three quantum charged particles and are widely used in bound state and scattering calculations. The approach is based on the spline collocation method and exploits intensively the tensor product form of discretized operators and preconditioner, which leads to a drastic economy in both computer resources and time.

Theoretical study of reactions in the three body $e^-e^+\bar{p}$ system and antihydrogen formation cross sections

“…We have proposed and implemented an approach for solving the quantum three-body problem which combines both a theoretically sound technique and a computationally efficient algorithm [10]. It is based on solving the Faddeev-Merkuriev (FM) equations [11] in total orbital momentum representation [12].…”

“…It is based on solving the Faddeev-Merkuriev (FM) equations [11] in total orbital momentum representation [12]. Our earlier calculations [10] have shown that our approach makes for highly precise calculations of the bound state energies for the states with high total orbital momentum. Here we apply it to scattering problems and perform a series of calculations of antihydrogen formation cross sections for the reaction (1).…”

“…The function F Lτ M M ′ is the common eigenfunction of the total orbital momentum squared, its projection and the spatial inversion operators [12,21] with eigenvalues L, M and τ . The multiplier (1 − z 2 α ) M ′ /2 in ( 11) is introduced to make the partial components ψ Lτ αM M ′ and their derivatives nonsingular at z α = ±1 [10,22]. Now substituting the series (11) into the FM equations (3) written in new coordinates (X α , Ω α ) and projecting the resulting equations onto the functions F Lτ M M ′ , one gets the finite set of 3D equations for partial components ψ Lτ αM M ′ (X α )…”

“…The obtained boundary problem is solved numerically by the spline collocation method. The scheme is described in [10] where the interested reader can find the details. For implementing the outgoing boundary conditions we use a hybrid basis, which is obtained by explicitly adding the outgoing waves to the spline basis set in variable y α .…”